# Non radical intervals with musical significance

 Ratio Cents Name Musical Significance $2^{1/\phi} \approx 1.5348$ 741.64 "Logarithmic phi" which divides the octave into two parts, one being Phi times larger than the other in cents. $\dfrac{\sqrt{5}+1}{2} \approx 1.6180$ 833.09 $\text{Phi } (\phi)$ "Linear phi," the unique interval whose continued fraction approximations converge more slowly than any other number. This is due to the continued fraction representation only containing 1's, as well as a general consequence of Dirichlet's Approximation Theorem. $e \approx 2.7183$ 1731.23 "e-tave" In Gene's black magic formulas, it is mathematically more "natural" to consider the number of divisions to the "e-tave" rather than the octave. $e^{2\pi} \approx 535.4917$ 10877.66 The zeta function has units that are given as divisions of the interval $e^{2\pi}$